Continuity of fuzzy approximate reasoning and its application to optimization

Abstract

This paper describes a mathematical framework for studying a nonlinear feedback control. The fuzzy control discussed here is the nonlinear feedback control in which the feedback laws are determined by IF-THEN type fuzzy production rules through approximate reasoning introduced by Nakamori. To prove existence of optimal control, we applied compactness of a set of membership functions in L2 space and continuity of the approximate reasoning, and prepared some propositions concerning approximate reasoning of Nakamori model. By considering fuzzy optimal control problems as problems of finding the minimum (maximum) value of the integral cost (benefit) function on an appropriate set of membership functions, the existence of fuzzy optimal control is shown. © Springer-Verlag Berlin Heidelberg 2007.